Chapter 10
EGR 272 – Circuit Theory II
1
Read: Chapter 10 in Electric Circuits, 9th Edition by Nilsson
Chapter 10 - Power Calculations in AC Circuits
Instantaneous Power:
p(t) = v(t)i(t) = instantaneous power
Average Power:
• Instantaneous power is not commonly used.
• Average power, P, is more useful.
• P = PAVG = PDC = average or real power (in watts, W)
• P can be defined for any waveform as:
P 
• For periodic waveforms with period T
(including sinusoids), P can be expressed as:
lim
1
p(t)dt

2T
T
P 
T
-T
1
T
p(t)dt

T
0
1
EGR 272 – Circuit Theory II
Chapter 10
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Average power can be calculated:
1) by inspection (for simple cases)
2) by integration (for more complex cases)
Example: Find the average power absorbed to the resistor below.
i(t)
i(t) [A]
+
v(t)
10
10
4
_
t [s]
0
2
4
6
8
10
12
EGR 272 – Circuit Theory II
Chapter 10
3
Example: Find the average power absorbed to the resistor below.
v(t) [V]
i(t)
+
v(t)
_
20
10
0
t [s]
0
2
4
6
8
10
12
Chapter 10
EGR 272 – Circuit Theory II
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RMS Voltage and Current
VRMS = root-mean-square voltage (also sometimes called Veff = effective
voltage)
VRMS = the square root of the average value of the function squared
VR M S 
1
T
T
 v (t)dt
2
0
I RMS 
1
T
T
 i (t)dt
2
0
Note how the name RMS essentially gives the definition.
Chapter 10
EGR 272 – Circuit Theory II
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Calculating average power for a resistive load using RMS values
A key reason that RMS values are used commonly in AC circuits is that they
are used in power calculations that are very similar to those used in DC circuits.
Recall that power in DC circuits can be calculated using:
P  V I 
V
2
 I R
2
R
Similarly, instantaneous power to a resistor can be calculated using
2
p(t)  v(t)  i(t) 
v (t)
 i (t)  R
2
R
Show that power can be also be calculated using RMS values as follows:
P  VR M S  I R M S
Development:

 VR M S 
R
2

 I RMS 
2
R
Chapter 10
EGR 272 – Circuit Theory II
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RMS values can be calculated:
1) by inspection (for simple cases)
2) by integration (for more complex cases)
Example: Find IRMS for the waveform below and use IRMS to calculate the
average power absorbed to the resistor. Recall that P was calculated earlier
using instantaneous power.
i(t)
i(t) [A]
+
v(t)
10
10
4
_
t [s]
0
2
4
6
8
10
12
EGR 272 – Circuit Theory II
Chapter 10
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Example: Find VRMS for the waveform below and use VRMS to calculate the
average power absorbed to the resistor. Recall that P was calculated earlier
using instantaneous power.
v(t) [V]
i(t)
+
v(t)
_
20
10
0
t [s]
0
2
4
6
8
10
12
Chapter 10
EGR 272 – Circuit Theory II
8
RMS value of sinusoidal voltages and currents
Using the definition of RMS voltage and current, it can be shown that for a
sinusoidal voltage v(t) = Vpcos(wt) and a sinusoidal current i(t) = Ipcos(wt) that:
VR M S 
Vp
2
 0.707V p
I RMS 
Ip
2
 0.707I p
Development: Derive the expression for VRMS above.
Example: Find the power absorbed by a 10  resistor with v(t) = 20cos(377t) V.
Chapter 10
EGR 272 – Circuit Theory II
9
Superposition of Power
Recall that, in general, superposition applies to voltage and current, but not to
power. i
Applying superposition to find the current i:
R
i = i1 + i 2
or
+
+
V
V
2
_
_
1
i = current due to V1 + current due to V2
Show that this leads to: P  P + P
1
Development:
P
1
T
T
 p(t)dt 
0
1
T
T
 i Rdt 
2
0
2
Chapter 10
EGR 272 – Circuit Theory II
10
Power to a Complex Load

If v(t) = Vm cos(wt) then i(t) = Im cos(wt - )  since
i(t)
+
Useful identities:
Show that:
P 
Vm  I m
2
cos (  )  V R M S  I R M S cos (  ) 
Development:
P
1
T
p(t)dt

T
0

1
T
v(t)i(t)dt

T
0

Vm Ð 0
ZÐ 
Z

 ImÐ   

cos(wt - ) = cos()cos(wt) + sin()sin(wt)
cos2 (wt) = ½ + ½ cos(2wt)
sin(wt)cos(wt) = ½ sin(2wt)
ZÐ°
v(t)
_
I=
V

 VR M S 
Z
2
cos (  ) 
 I RMS 
2
 Z cos (  )
Chapter 10
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Key definitions:
Recall that real or average power, P, was just defined as:
P 
Vm  I m
2
cos (  )  V R M S  I R M S cos (  ) 
 VR M S 
2
cos (  ) 
Z
 I RMS 
2
 Z cos (  )
Other important terms are apparent power and power factor, as defined below:
Apparent power = (VRMS)(IRMS) measured in volt-amperes, VA
power factor  p.f.  Fp 
real power
 cos(  )
apparent power
Since cos is an even function (i.e., cos() = cos(-)), calculating cos() does not
reveal the sign of the angle, so the terms leading and lagging are used with
power factor.
• The power factor is leading if  < 0, (i.e., I leads V or the load is capacitive).
• The power factor is lagging if  > 0, (i.e., I lags V or the load is inductive).
Chapter 10
EGR 272 – Circuit Theory II
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Examples: Determine P and the p.f. for each case below:
+
10V RMS
_
Z
a)
Z  4
b)
Z  j3
d)
Z  4  j3
e)
Z  4 - j3
c)
Z  -j3
Chapter 10
EGR 272 – Circuit Theory II
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Complex Power:
If v(t)  V m cos (w t),
then V  V m Ð 0 (pha sor using the peak value of v(t))
but w e can also define V R M S  V R M S Ð 0
(phasor using the R M S value of v(t))
The use of phasors with RMS values is very convenient for power calculations.
IRMS
V R M S  VR M S Ð 0
+
VRMS
Z
since
_
Define
I RMS =
S  com plex pow er 
im plies that I R M S  I R M S Ð  
VR M S

VR M S Ð 0
ZÐ 
Z
V RMS  I RMS
*
 I RMSÐ  
(in volt-am peres, V A)
Chapter 10
Since
EGR 272 – Circuit Theory II
S  com plex pow er 
14
V RMS  I RMS
Show that S  S Ð    VRM S  I RM S Ð  
where
*

(in volt-am peres, V A)
P  jQ
P  real or average power  V RM S  I RM S cos(  )
Q  reactive power  V RM S  I RM Ssin(  )
S
 apparent pow er  V R M S  I R M S 
p.f.  pow er factor 
P
S
Development:
 cos(  )
P
2
(in watts, W )
(in volt-amperes reactive, VAR)
 Q
2
(in volt-am peres reactive, V A R )
(expre ssed as leading if   0 or lagging if   0 )
Chapter 10
EGR 272 – Circuit Theory II
15
Complex Power Diagram
A complex power diagram is sometimes used to illustrate the relationship
Im
between P, Q, |S|, and S.
S
Q
|S|
or
|S|
Q


P
Re
P
Example:
If VRMS = 100Ð0° V and IRMS = 5 Ð-30° A, find P, Q, |S|, and S and illustrate
them on a complex power diagram. Also find the power factor.
EGR 272 – Circuit Theory II
Chapter 10
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Other useful relationships
Substitute VRMS = ZIRMS and Z = R + jX into S = VRMSIRMS* to show that:
PR 
 VR M S 
2

R
Q L (or Q C ) 
 I RMS 
 VR M S 
X
Development:
2
and PL  PC  0
R
2

 I RMS 
2
X
and
QR  0
-1 

w
here
X

w
L
and
X

L
C


wC 

Chapter 10
EGR 272 – Circuit Theory II
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Two approaches for performing power calculations:
1. Find Z T otal and I R M S  V R M S
T hen find S  V R M S  I R M S
Z T otal
*
 P  jQ
2. Find the voltage or current for each component (magnitude only). Then
calculate P or Q for each component and add the results for total P and Q.
T hen use PT 
P
com ponent
,
Q
T

Q
com ponent
, and
S T  PT  jQ T
EGR 272 – Circuit Theory II
Chapter 10
18
Example: Find S, S , P, Q, and p.f.
A) By first finding the total circuit impedance
+
100V RMS
60 Hz
_
10
15
26.5 mH
132.6 uF
Chapter 10
EGR 272 – Circuit Theory II
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Example: (continued) Find S, S , P, Q, and p.f.
B) By finding P or Q for each component (use mesh equations first to find the
currents for each component).
KVL, mesh 1 : - 100 Ð 0 °  10 I 1  j10( I 1 - I 2 )  0
+
100V RMS
60 Hz
_
10
KVL, mesh 2 : j10( I 2 - I 1 )  15 I 2 - j20 I 2  0
15
26.5 mH
132.6 uF
1 0  j10

 - j10
- j10   I 1  100 
   

15 - j10   I 2   0 
I 1  5.099 Ð - 41.8 ° A
I 2  2.828 Ð 81 . 9 ° A
I L  I 1 - I 2  7.071 Ð - 61.2 ° A
Chapter 10
EGR 272 – Circuit Theory II
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Power Factor Correction
Power companies charge their customers based on the real power (P) used. If
the customer has a significant amount of reactive power (or a low power factor)
this has no effect on P, but results in higher current levels. Power companies
“encourage” their large customers to “correct” their power factor. If their
power factor drops too low (perhaps less than 0.9), the power company may
charge higher rates.
Large companies often have lagging power factors due to large amounts of
machinery (inductive loads) and they can correct their power factor by adding
in large parallel capacitors (or use synchronous motors). This generates
negative reactive power that cancels the positive reactive power due to the
inductive loads resulting in a power factor which is near or close to unity. The
corrected power factor results in lower current levels and thus lower line losses
(I2 R) as the power company delivers the power to the customer. Use user also
benefits from lower current levels since maximum current ratings on wire and
equipment can be reduced.
Chapter 10
EGR 272 – Circuit Theory II
21
Illustration - Power Factor Correction
I1
I2
+
I1
+
IC
C
Circuit
_
Circuit
_
Original Circuit
New Circuit (Note that |I2| < |I1| )
Im
Im
Im
Q
Re
Re
Re
P
P
QC = -Q
Original Circuit
S = P + jQ
+
Added Capacitor
S = P + jQ
=
New Circuit
S = P + jQ - jQ = P
Chapter 10
EGR 272 – Circuit Theory II
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Example: Add a capacitor in parallel with the source in order to:
A) correct the power factor to unity
Also calculate the total source current and compare it to the original current.
Note that this is the same circuit analyzed previously.
+
100V RMS
60 Hz
_
10
15
26.5 mH
132.6 uF
Chapter 10
EGR 272 – Circuit Theory II
23
Example: (continued) Add a capacitor in parallel with the source in order to:
B) correct the power factor to 0.95, lagging
Also calculate the total source current and compare it to the original current.
Summary: (Parts A and B)
C ircuit
p.f.
O riginal
C (added)
---
P art A
1.0 (unity )
P art B
0.95, lagging
current, |I|
R eduction in |I|
---
Chapter 10
EGR 272 – Circuit Theory II
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Power Factor in Systems
Just as total power can be determined by adding the power dissipated by each
component, so can the total power for a system be determined by adding the
power dissipated by each subsystem.
T hen use PT otal 
P
subsystem
,
Q
T otal

Q
subsystem
, and
S T  PT  jQ T
Example: Find the power factor of the entire system below that is comprised
of three subsystems as shown below.
+
_
Subsystem 1:
|S| = 2000 VA
p.f. = 0.8, lagging
Subsystem 2:
P = 2000 W
p.f. = 0.9, leading
Subsystem 3:
|S| = 2000 VA
Q = 1500 VAR
EGR 272 – Circuit Theory II
Chapter 10
25
Transformers:
Our earlier study of inductors introduced the idea that current through a set of
windings creates a magnetic field and magnetic flux, , flows through the core.
If  then flows through a second set of windings another magnetic field is
created and a secondary voltage is induced across these windings. This
arrangement of two windings on a common core is called a transformer
(illustrated below).

i1
i2
+
+
V1
N1
turns
_
N2
turns
V2
_
Primary
windings
Secondary
windings
Load
EGR 272 – Circuit Theory II
Chapter 10
Recall that
 = flux linkage
N = number of turns
 = magnetic flux (in Webers, Wb)
and  = N = Li , where L = inductance in Henries, H
Also
d
v 

d(N  )
dt
So
v  N
dt
d
dt
Since the same magnetic flux, , flows through the windings it is also true that
the rate of change of magnetic flux, d/dt is the same, so
d

dt
a 
V1

N1
N1
N2
V2
N2

V1
V2
,
so
N1
N2

V1
V2
where a = turns ratio
26
EGR 272 – Circuit Theory II
Chapter 10
27
Key transformer relationships
We just saw that
Similarly
N1
a 
a 

N2
V2
N1
I2

N2
If
Z1 
V1
I1
and
Z2

V1
I1
V2
I2
where a = turns ratio
so
then sh ow that
a 
N1

N2
a
2

V1
V2
Z1
Z2

I2
I1
Chapter 10
EGR 272 – Circuit Theory II
Transformer symbols
General Transformer
Iron-core Transformer
28
Chapter 10
EGR 272 – Circuit Theory II
29
Examples of transformers (reference: www.allelectronics.com)
Primary: 120V
Secondary: 28V, 1.5A
Primary: 110V
Secondary: 15V, 0.4A
Primary: 120V
Secondary: 40VCT, 0.25A
Utility pole
transformer
Chapter 10
EGR 272 – Circuit Theory II
30
Examples of transformers (reference: www.allelectronics.com)
PC Mount Transformer
Primary: 120V
Secondary: 16VCT, 0.8A or 8V, 1.6A
Up/Down Transformer
(110V to 220V) or (220V to 110V)
Toroidal Transformer
Primary: 120V
Secondary: 8.5V and 9.4V
Variac (Variable Transformer)
Input: 110V
Output: 0 to 130V
Chapter 10
EGR 272 – Circuit Theory II
31
Dot Convention
The direction of the windings in the secondary with respect to the direction of
the windings in the primary determines the polarity of the secondary voltage.
However, rather than showing the direction of the windings, dots are often
placed at one end of each winding. Then the following convention applies:
Dot convention: “A positive voltage at one dotted terminal
will produce a positive voltage at the other dotted terminal.”
Note that the relationships a  N 1  V1  I 2
N2
V2
I1
Imply that:
• the positive terminals of V1 and V2 are at the dotted terminals
• I1 enters the dotted terminal and I2 leaves the dotted terminal
a:1
N1:N2
i1
These voltage polarities and
current directions are indicated
to the right.
i2
+
+
V1
V2
_
_
Chapter 10
EGR 272 – Circuit Theory II
Four Common Transformer Uses
• Change voltage levels
• Change current levels
• Change impedance levels (impedance matching)
• Isolation (e.g., to isolate the secondary load from the ground in the primary)
Examples: Show simple examples of the four transformer uses listed above.
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EGR 272 – Circuit Theory II
Chapter 10
33
Example: Determine the V1 and i2 in the circuit below. Find these values by:
A) Using KVL equations around each loop
1:2
+
20 mF 5
+
2
V1
5 cos(10t)
_
_
20 mH
i2
EGR 272 – Circuit Theory II
Chapter 10
34
Example: Determine the V1 and i2 in the circuit below. Find these values by:
B) Reflecting the impedance of the secondary to the primary.
1:2
+
20 mF 5
+
2
V1
5 cos(10t)
_
_
20 mH
i2
EGR 272 – Circuit Theory II
Chapter 10
35
Example: Determine the value of Zab below.
1:2
a
10
Zab
3:1
10
1:2
10
10
b
Chapter 10
EGR 272 – Circuit Theory II
36
Example: Recall the example presented in EGR 271 where a transformer was
used to insure that maximum power was delivered to a 4 ohm speaker from the
output of an amplifier with an output resistance of 100 ohms. Illustrate and
discuss.
Chapter 10
EGR 272 – Circuit Theory II
37
Example: Determine the inductance, L, and the turns ratio, a, required to
achieve maximum power transfer to the secondary load.
a:1
+
62.5 pF
80 
320
L
100cos(105t) V
_
Chapter 10
EGR 272 – Circuit Theory II
Transformer Models:
Transformers considered in this
course have been considered to
be ideal. Although transformer
behavior is often close to ideal,
there are cases where non-ideal
behavior might be examined.
The figures to the left
(reference: Electric Power &
Machinery by Matsch) show
models of a transformer used to
approximate non-ideal behavior.
The resistances, R1 and R2,
represent the resistance of the
windings and the inductive
reactances, Xl1 and Xl2 represent
leakage magnetic flux.
38